On the symmetry of primes

We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say $I_{\Lambda}(N,h)$, of the von Mangoldt function $\Lambda(n)$ ($:= \log p$ for prime-powers $n=p^r$, 0 otherwise). We get $I_{\Lambda}(N,h)\ll NhL^5+Nh^{21/20}L^2$, with $L:=\log N$; as a Corollary, we bound non-trivially the Selberg integral of the primes, i.e. the mean-square of $\sum_{x<n\le x+h}\Lambda(n)-h$, over $x\in [N,2N]$, to get the "Prime Number Theorem in almost all short intervals" of (log-powers!) length $h\ge L^{11/2+\epsilon}$. We trust here in the improvement of the exponent, say $c<11/2$.